List of resolved prime conjectures

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This is a list of resolved prime conjectures.

Proven conjectures

The prime number theorem was a conjecture for almost a century. Carl Friedrich Gauß reportedly came up with the formula in the late 18th Century, but even he doubted himself and along with other mathematicians came up with formulas along the lines of $\pi (x)\approx {\frac {x}{\log(x)-A}}$. Jacques Hadamard and Charles Jean de la Vallée-Poussin proved the prime number theorem independently of each other in 1896 using complex analysis.

Bertrand's postulate was a conjecture for a much shorter time: Joseph Bertrand asserted this in 1845 and Pafnuty Chebyshev was the one who proved it in 1850.

Disproven conjectures

Disproven with a counterexample

Pólya's conjecture was posed by George Pólya in 1919, a counterexample was found in 1960.

The so-called Chinese hypothesis (the converse of Fermat's little theorem) could have been disproven much sooner than it was by the use of congruences to show that $2^{340}\equiv 1\mod 341$.

Disproven without a counterexample

For some conjectures there can't be a counterexample, such as if the conjecture asserts a given equation has solutions but in fact it doesn't. But even conjectures for which a counterexample may exist are sometimes disproven without actually finding a counterexample. This was the case with the aforementioned conjecture from Pólya: C. B. Haselgrove disproved it in 1958, but the first counterexample was found two years later by R. S. Lehman.

Standing conjectures

See list of prime conjectures.